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A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is ...
For saddle point problems, however, many discretizations are unstable, giving rise to artifacts such as spurious oscillations. The LBB condition gives criteria for when a discretization of a saddle point problem is stable. The condition is variously referred to as the LBB condition, the Babuška–Brezzi condition, or the "inf-sup" condition.
The two critical points occur at saddle points where x = 1 and x = −1. In order to solve this problem with a numerical optimization technique, we must first transform this problem such that the critical points occur at local minima. This is done by computing the magnitude of the gradient of the unconstrained optimization problem.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Another condition in which the min-max and max-min are equal is when the Lagrangian has a saddle point: (x∗, λ∗) is a saddle point of the Lagrange function L if and only if x∗ is an optimal solution to the primal, λ∗ is an optimal solution to the dual, and the optimal values in the indicated problems are equal to each other. [18 ...
The first is the only solution in the interior of the upper quadrant. It is a saddle point (as shown below). The second is a repelling point. The third is a degenerate stable equilibrium. The first solution is meant by default, although the other two are important to keep track of. Any optimal trajectory must follow the dynamical system.
One can ask whether a minimizer point of the original, constrained optimization problem (assuming one exists) has to satisfy the above KKT conditions. This is similar to asking under what conditions the minimizer x ∗ {\displaystyle x^{*}} of a function f ( x ) {\displaystyle f(x)} in an unconstrained problem has to satisfy the condition ∇ f ...